Combination Calculator (nCr)

Understanding Combinations

What is a Combination?

A combination is a way of selecting items from a collection, where the order of selection does not matter. It answers the question: "How many ways can we choose a subset of items from a larger set, regardless of order?"

The Combination Formula

The formula for combinations is:

\[C(n,r) = \frac{n!}{r!(n-r)!}\]

Where:

• n = total number of items to choose from
• r = number of items being chosen
• ! denotes factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1)

Calculation Steps

To calculate a combination:

1. Identify n and r
2. Calculate n!
3. Calculate r!
4. Calculate (n-r)!
5. Apply the formula: n! / (r! * (n-r)!)

Example Calculation

Let's calculate 5C3 (combination of 5 items taken 3 at a time)

\[C(5,3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{5 \times 4 \times 3!}{3! \times 2 \times 1} = \frac{20}{2} = 10\]

Visual Representation

This diagram illustrates the calculation of 5C3, showing how we choose 3 items from a set of 5, regardless of order.